I am reading Dummit and Foote (D&F) Section 13.1 Basic Theory of Field Extensions.

I have a question regarding the nature of extension fields.

Theorem 4 (D&F Section 13.1, page 513) states the following (see attachment):

---------------------------------------------------------------------------------------------------------------

Theorem 4. Let be an irreducible polynomial of degree n over a field F and let K be the field . Let . Then the elements

are a basis for K as a vector space over F, so the degree of the extension is n i.e.

. Hence

consists of all polynomials of degree in

------------------------------------------------------------------------------------------------------------------------------

However, when we come to Example 4 on page 515 of D&F we read the following: (see attachment)

(4) Let and which is irreducible by Eisenstein.

Denoting a root of p(x) by we obtain the field

with an extension of degree 3. ... ... etc

------------------------------------------------------------------------------------------------------------------------------

Now my problem is that in Theorem 4 we read

which becomes

in the situation of Example 4

But then in Example 4 we have

???

It seems that in Theorem 4, we have but in Example (4) we have and we do not have equality but only an isomorphism, that is .

In Field theory we seem to prove that an irreducible polynomial has a root in a field that is isomorphic to the actual field that contains the root.

Does what I am saying make sense? Can someone clarify this issue for me?

Peter